The one-dimensional discrete cosine transform is similar to the two-dimensional transform, except that we drop the second variable ( j or y) and the second cosine factor. We also drop, from the inverse DCT only, the leading 1/ √ 2N coefficient. Implement this and its inverse for N = 8 (a spreadsheet will do, although a language supporting matrices might be better) and answer the following:
(a) If the input data is {1, 2, 3, 5, 5, 3, 2, 1}, which DCT coefficients are near 0?
(b) If the data is {1, 2, 3, 4, 5, 6, 7, 8}, how many DCT coefficients must we keep so that after the inverse DCT the values are all within 1% of their original values? 10%? Assume dropped DCT coefficients are replaced with 0s.
(c) Let si , for 1 ≤ i ≤ 8, be the input sequence consisting of a 1 in position i and 0 in position j, j
= i. Suppose we apply the DCT to si , zero the last three coefficients, and then apply the inverse DCT. Which i, 1 ≤ i ≤ 8, results in the smallest error in the ith place in the result? The largest error?